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   "source": [
    "# [Integer Right Triangles](https://projecteuler.net/problem=39)\n",
    "\n",
    "If a right triangle has integer side lengths, the side lengths are a [Pythagorean triple](https://en.wikipedia.org/wiki/Pythagorean_triple). In [problem 9](https://projecteuler.net/problem=9), we wrote a generator for primitive Pythagorean triples based off of Euclid's formula. We can modify that generator to cut off after the triplets have passed a maximum perimeter. Note that a triangle with side lengths generated by Euclid's formula will have perimeter $2m^2 + 2mn$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "b03eb872",
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   "source": [
    "from itertools import count\n",
    "\n",
    "def primitive_pythagorean_triplets(max_perim):\n",
    "    for m in count(2):\n",
    "        if 2*m^2 + 2*m > max_perim:\n",
    "            break\n",
    "\n",
    "        for n in range(1, m):\n",
    "            if not ((m % 2) != (n % 2)) or gcd(m, n) != 1:\n",
    "                continue\n",
    "            \n",
    "            a = m^2 - n^2\n",
    "            b = 2*m*n\n",
    "            c = m^2 + n^2\n",
    "            \n",
    "            if a + b + c > max_perim:\n",
    "                break\n",
    "            \n",
    "            yield (a, b, c)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "431ef21c",
   "metadata": {},
   "source": [
    "Now we can just iterate through our new generator and group each triangle by their perimeters. We also multiply to consider non-primitive triplets."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "62aa955f",
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "max_perim = 1000\n",
    "perimeters = dict()\n",
    "for (a, b, c) in primitive_pythagorean_triplets(max_perim):\n",
    "    for k in count(1):\n",
    "        perimeter = k * (a + b + c)\n",
    "        if perimeter > max_perim:\n",
    "            break\n",
    "        \n",
    "        if perimeter not in perimeters:\n",
    "            perimeters[perimeter] = set()\n",
    "        perimeters[perimeter].add((k*a, k*b, k*c))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f776a5c6",
   "metadata": {},
   "source": [
    "Our answer is whichever perimeter has the highest total."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "0cd6241f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "840"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = max(perimeters, key=lambda x: len(perimeters[x]))\n",
    "p"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f32fb164",
   "metadata": {},
   "source": [
    "There are eight right triangles with this perimeter."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "9c31c68f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(105, 360, 375),\n",
       " (140, 336, 364),\n",
       " (210, 280, 350),\n",
       " (252, 240, 348),\n",
       " (315, 168, 357),\n",
       " (350, 120, 370),\n",
       " (390, 56, 394),\n",
       " (399, 40, 401)}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "perimeters[p]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6cb1b692",
   "metadata": {},
   "source": [
    "## Related sequences\n",
    "* Number of integer right triangles with perimeter $n$: [A024155](https://oeis.org/A024155)\n",
    "\n",
    "#### Copyright (C) 2025 filifa\n",
    "\n",
    "This work is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International license](https://creativecommons.org/licenses/by-sa/4.0/) and the [BSD Zero Clause license](https://spdx.org/licenses/0BSD.html)."
   ]
  }
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